One way in which traditional storytelling differs from Modernist stories is that Modernist stories focus on the individual's experience, whereas traditional stories focused on an objective story.
Modernists did not believe in absolute truth -- they believed the truth varied from person to person. Therefore, Modernist stories are more likely to be told from a first-person perspective. The use of a stream-of-consciousness technique is also common in Modernist literature, as this technique takes the reader into the mind of a character so that the reader can experience the character's subjective reality.
Traditional stories, on the other hand, often had an objective narrator describing the events of the novel, events which often had a clearly definied beginning, middle, and end. Because Modernists believed life was not so orderly, their stories did not feature this traditional plot structure.